Pyramids and Cones Volume

Key Concepts

• Find volume of a composite solid.
• Solve a multi-step problem.

Introduction

Relation between the volume of a cone and a cylinder

The contents of three conical cups exactly fill a cylinder of the same radius and height. 

Relation between the volume of a cone and a cylinder

The contents of three pyramids with rectangular bases exactly fill a prism of the same base and height. 

Find volume of a composite solid 

Example 1: 

Find the volume of the solid shown. 

Solution: 

V (solid) = V(small) + V(large) 

=1/3 π (5)²7 + 1/3 π (5)²11  

=25/3π (7+11) 

=25/3 π (18) 

=150  π  

≈ 471.2 in³ 

The volume of the solid is 471.2 in³. 

Example 2: 

Find the volume of the solid shown. 

Solution: 

V (solid) = v(prism) + v(pyramid) 

= Bh + 1/3 Bh₁ 

= 20.4 × 12 × 10 + 1/3 (20.4 × 12)9.1 

= (20.4 × 12)(10 +9.1/ 3) 

≈ 3190.6 m³ 

The volume of the solid is 3190.6 m³. 

Solve a multi-step problem 

Example 3: 

John is building a music studio in his backyard. To buy a heating unit for the space in his studio, he needs to determine the BTUs (British Thermal Units) required to heat the space. He found that there should be 2 BTUs per cubic foot for new construction with good insulation. What size unit does John need to purchase in order to heat the studio? 

Solution: 

The building of the studio can be broken down into two shapes, that is, the rectangular base and the pyramid ceiling. The volume of the base can be calculated as, 

V = l × w × h 

    = (25)(25)(8) 

    = 5000 ft³ 

The volume of the ceiling can be calculated as, 

V = 1/3 Bh 

 = 1/3 (25)(25)(8) 

 = 1666.67 ft³ 

The total volume is therefore 5000 + 1666.67 = 6666.67 ft³. Two BTU’s are needed for every cubic foot, so the size of the heating unit John should buy is 6666.67 × 2 = 13,333 BTUs. 

Example 4: 

A rocket has the dimensions as shown below. If 62% of the space in the rocket is needed for fuel, what is the volume of the portion of the rocket that is available for filling nonfuel items? Round your answer to the nearest cubic inch. 

Solution: 

STEP 1: Find the combined volume of the rocket that is,  

Volume of the cylinder = Bh 

V = πr²h 

 = π x 6² x 38 

 = 1368π 

Volume of the cone = 1/3 Bh 

V = 1/3 x πr²h 

    = 1/3 x π6² x 8 

    = 96π 

Total volume = 1464π  

≈ 4599.3 

STEP 2: Find the available space for filling nonfuel items, 

38% of 4599.3 

= 38/100 x 4599.3 

= 1748 

The available space for filling nonfuel items is 1748 inch³. 

Exercise

1. ________________________ is a combination of any two shapes.
2. A right cylinder with a radius of 3 centimeters and a height of 10 centimeters has a right cone on top of it with the same base and a height of 5 centimeters. Find the volume of the solid. Round your answer to two decimal places.
3. Find the volume of the solid shown. Round your answer to two decimal places.

4. Find the volume of the solid shown. Round your answer to two decimal places.

5. Find the volume of the solid shown. Round your answer to two decimal places.

6. Find the volume of the solid shown. Round your answer to two decimal places.

7. A pastry bag filled with frosting has a height of 12 inches and a radius of 4 inches. A cake decorator can make 15 flowers using one bag of frosting.
   • How much frosting is in the pastry bag?
     • Round your answer to the nearest cubic inch.
   • How many cubic inches of frosting is used to make each flower?

8. A snack stand serves a small order of popcorn in a cone-shaped cup and a large order of popcorn in a cylindrical cup.

How many small cups of popcorn do you have to buy to equal the amount of popcorn in a large container? Explain.

9. Which container gives you more popcorn for your money? Explain.

10. Assume that the automatic pet feeder is a right cylinder on top of a right cone of the same radius. (1 cup 5 14.4 in.3)
    • Calculate the amount of food in cups that can be placed in the feeder.
    • A cat eats one-third of a cup of food twice per day. How many days will the feeder have food without refilling it?

Concept Map

What have we learned

• Find the volume of a composite solid by adding the volumes of each shape.
• Solve a multi-step problem.